Microwave filter



Dec. 23, 1952 o. J. ZOBEL 2,623,120

MICROWAVE FILTER Filed April 20, 1950 s Sheets-Sheet 2 FIG. FIG. /2

1x 12, 47 37 as 42 45 39 40 43 4a A' g,

12x I 27 25w 2 25 27 NORMA L IZED MM 65 IMPE DANCE FIG. /6 24 28 lNVENTOR 0. J 2085 L BY 2m 0/25 WfW A 7' TORNEY Patented Dec. 23 1952 MIGHOWAVE FILTER Otto J. Zobel, Morristown, N. J.,.assignor to Bell Telephone Laboratories, Incorporated, New York,,N. 3L, a-eorporation of New York Application April- 20, 1950, Serial N 0. 156,959-

40 Claims.

This. invention: relates to wave. transmission.

networks and moreparticularlyyto microwave filters and impedance transformers.

An object of the; invention isito providea microwave filter in which the width and the locationof. the transmission band may be preassigned. at will.

Another object is to increase therelative-band crowave impedance transformer which may have.

any desiredtransformation ratio.

A further object is to providecsuchz transformer having image. impedances. suitable; for connectinga microwave filterto: a waveguide.-

The microwave filters and. transformers of the present invention are composite structures made.

up of individual sectionswhich' are designed on the basis of their image parameters, that is, the

transfer constant andimage impedances. Structurally, the network comprises a hollow-pipe wave guiderof'uniformcross: section with shunt im-- pedances; either inductive or capacitive, located at spaced intervals. Theseimpedances take the form of partial obstructions'across thewave guideand may, for example, be apertured transverse partitions, often called irises.

The network sections may be either'of the midguide type, constituted by a'section of Wave guide with acentrally positionediris, or the mid-iris type, made upof two likevirisesatthe respective 1 ends of a section of ,wave guide. Such a section has a transmission band the limits of which are determined by the length of *theguide and the impedance of the iris, Composite filter structures are built up by connecting in tandem'any num-'- ber of such sections, usually. all of the same type, having the same imagegimpedance andthe same transmission band. The band width obtainable is limited only by the transmission" characteristic of the component wave guide. Any desired-dis=-- crimination maybeobtained by providing a Suificient number of sections:

The transformers are made up or two or more tandem sections which differ inimage impedance. The'sections may be all of .the same type or they may be mixed in type. Thelocationof the transmission band may be chosen at will. Any desired transformation ratio may be obtained.

The transformer has image impedances which are well suited .to;matching .a microwave filter of the type. describedabove'to: aiwave Lgui'de. An

important combination in accordance with. the

invention is a wave filter, made up of any num-*- ber of like mid-guide or mid-iris sections, with a transformer at each end for matching the imageimpedance of the filter to the' characteristic 1m-" pedance of the wave guide used in the filter sec tions. In this structure the Wave guide used in the filter and the transformers may be of the same cross section as that of the input and output wave guides, thus simplifyinggthe connections.

The: filter and the transformers Will ordinarily.

have substantially the same transmissionband; The impedance match may be made .very close over .a wide band, thus assuring. a very. fiat in'-. sertion loss characteristic throughoutthe band". Two such transformers connected back-to-:ba'ck,. with the central filter sections omitted, .also pro-: vide a goodband-pass characteristic.

The nature of. the invention will benmore fully. understood from the following detailed description and by reference to the accompanyingsdraw-- ings, of which:

Figs. 1 and 2 are schematic representations, respectively, of a mid-guide and a mid-iris wave filter section;

Figs. 3 and 5 show, respectively, an'inducti-ve iris and'a capacitive iris in a wave guide of;rectangular cross section;

Figs..4 and 6 show phase-frequency. character; istics associated, respectively, with" an inductive iris and a capacitive iris, referred to in explaining ing the invention;

Fig. '7 shows schematically a band-pass impedance transformer in accordance with' the 1m vention comprising two dissimilar mid -g-llide'TSeG-= tions connected in tandem;

Fig. 8 represents a similarly-constituted transformer made up of two dissimilar mid-iris sections;

Figs. 9 and 10 show B-sectiontransformers in which the sections are, respectively, of the midguide type and the mid-iris type;

Figs. 11 and 12' represent -section transformers made up of mid-guide and mid-iris sec tions,'respectively;

Figs. 13 to 16 represent transformers made up" of tandem connected, mixed combinations of dissimilar sections, these being' in Fig. 13 a midguide and a mid-iris, in Fig. 14 two mid-guideand amid-iris, in Fig. 15 amid-guideand two mid-iris, and in Fig. 16 two mid-guide and-two mid-iris sections;

Fig. 17 is a schematic circuit of a networkcom prising a 2-section transformer and. a filter-sec tion, in which all of the sections are-of the-mid guide type and the associated shunt impedances are inductive;

Fig. 18 gives the normalized image impedancefrequency characteristics of the wave guide, the filter, and the transformer sections shown in Fig. 17, and also the characteristic of the transformer sections in combination;

Fig. 19 is a schematic circuit of a network made up of two networks of the type shown in Fig. 17, connected back-to-back, with two additional filter sections of the same type inserted between them;

Fig. 20 is a longitudinal sectional view of a physical embodiment of the network shown in Fig. 19;

Fig. 21 is a transverse sectional view of the structure shown in Fig. 20, as seen at the plane 2l2l looking in the direction of the arrows;

Fig. 22 gives the insertion loss characteristic of the network of Fig. 20, between wave guide, and for comparison the corresponding characteristic of the four filter sections; and

Fig. 23 gives the calculated and measured insertion loss characteristics of the two transformers of Fig. 20 connected back-to-back, with the filter sections omitted.

Properties of wave guides and irises Before deriving the design formulas applicable to the microwave filter and transformer structures of the present invention, the properties of the associated wave guide itself and of irises will first be briefly considered. In the following discussion it will be assumed that the wave guide is a hollow metallic pipe of uniform, rectangular cross section and is transmitting electromagnetic waves of the TE1,0 mode. Such a guide 50 is shown in cross section in Fig. 3. The transverse dimensions a and b are unequal, a usually being approximately equal to 2b. The electric field E,

the direction of which is indicated by the arrow, is perpendicular to the longer dimension a. It is to be understood, however, that, with appropriate modifications, the design method is applicable to wave guides having other cross-sectional shapes, for example, circular.

A guide such as 50 transmits electromagnetic waves selectively like a high-pass filter but in various modes. For the TE1,0 mode the critical or cut-ofi frequency fc is given by the expression where c is the velocity of light and is equal to 3x10 centimeters per second. At this frequency the critical wavelength \c is equal to 2a.

The propagation constant, 7, above fc is =a+ip (2) where the attenuation constant, a, is here negligible and the phase constant, p, at any frequency, f, s

The characteristic impedance, is, may be written where its is the characteristic impedance at infinite frequency. Thus, for design purposes the wave guide can be treated as a non-dissipative transmission line having the propagation constant and characteristic impedance given above.

When an iris, or thin apertured metal diaphragm, is inserted transversely across a wave General periodic structure The general periodic structure under consideration comprises a uniform wave guide with identical irises, each of impedance iX, inserted within the guide at regular intervals with a spacing of 21 between irises. For purposes of analysis the structure may be divided into identical minimum symmetrical sections. If the points of division are taken at the mid-points of the connecting lengths of guide, the sections are of the mid-guide type shown schematically in Fig. 1. Such a section comprises a section of wave guide of length 22 with an iris of impedance iX connected in shunt at the center. The image impedance at each end of the mid-guide section is denoted by W. Alternatively, if the structure is sectioned at the mid-points of the irises, a mid-iris section, shown in Fig. 2, is obtained. The section consists of two irises, each of impedance i2X, separated by a section of wave guide of length 2Z. The image impedance at each end is W.

The simplest method of deriving formulas for W and W, and the transfer constant T of either section, is to consider a mid-half section, obtained by dividing either the mid-guide or midiris section at its center. The mid-half section thus comprises a section of wave guide of length Z with an iris of impedance i2X at one end thereof. It is an unsymmetrical structure with an image impedance W at the mid-guide end and an image impedance W at the mid-iris end. The transfer constant of the mid-half section is half that of a full section, T, given by T=A+iB where A is the attenuation constant and B is the phase constant of the full section. From standard formulas for the open-circuit and short-circult driving-point impedances we arrive at the relation 1 tanh T tan W (5) where 6 is an angle determined by the iris impedance in accordance with the formula tan 6E2X/lc (6) The normalized mi-guide image impedance w, obtained by dividing the image impedance W by the characteristic impedance is of the associated wave guide, is given by the expression Similarly, the normalized mid-iris image impedance, w, is

These two impedances are simply related.

Wave-guide wave filter with inductive irises Fig. 3 shows one form of inductive iris suitable for use in the networks of the invention. The

rectangular wave guide 50 with unequal transverse dimensions aand b is shown cross sectioned just ahead of the iris which is formed by the two halves 5|, 52 of a thin, metallic, transverse partition with a central aperture 53 therebetween. The aperture 53 extends all the way between the wider sides of the guide 50 in the direction of the electric field E and has a width (1 in the direction of the dimension a. It is to be understood, however, that the inductive iris may take other physical forms. For example, the aperture 53 may be circular or of other suitable shape.

Application of the above formulas will be made to the design of a wave filter with inductive irises of the type shown in Fig. 3, so that there will be a preassigned transmitting band above is between the cut-off frequencies i1 and f2 which is not limited in width. For a given wave guide, f1 and f2 determine the two parameters of this structure, that is, the magnitude and spacing of the irises. This is the first designproblem arising here.

The normalized susceptance of an inductive iris will be denoted by 5;, where the subscript 2' means inductive. Similarly, when the subscript 2' is applied to the symbols X, 6, Z, A, T, B, w, and w it will have the same meaning. Since s1 is negative and proportional to the wavelength, A in the wave guide, we may put where i :21r/p and g is a positive design constant fixing the magnitude of the susceptance. Then which has the frequency characteristic of an inductive reactance. Substitution in the general formula 6 gives which shows that 61 is positive with the inductive 11'15.

The selective properties of the wave filter can be visualized from the phase-frequency characteristics of ,sli-i-ti, given by curve 54 of Fig. 4, and 511, shown as curve 55. In the transmitting band where the attenuation constant A1 0, Tizz'Bi, an and wi are real, the functions tan (,BZi-f-ti) and tan 5Z1 are of opposite signs. At the preassigned limits of f1 and f2 of the lowest band under consideration these tangents are, respectively, infinite. At fiq, defined here as the quarter-wave frequency where the phase constant of the section has the value Biq:7T/2 radians, tan (cm-6i) and tan ,Bli are equal and opposite. Hence,

at ,fi, fi1Zi:1r/2-5u (13) at fiq, Bqli:1r/25ir1/ (15) Biq:7r/ 2 (16) and at f2, BZZi 'Ir/Z (17) Bi2:'n' (18) In Equations 13 and 14 the subscript 1 used with the symbols B, B, and 6 indicates that the value of the angle at the frequency h is to be taken, and similarly in Equations 17 and 18 the subscript 2 associated with B and indicates the value at f2. In Equations 15 and 16 the subscript q means that the value of the angle at the frequency fiq is to be used.

For convenience, I shall define two radian angles which arise here with inductive irises,

namely.

(f /f=) (firm- (19) and iq c 2-1 Then 2fi Zi: pi:1r-6i (21) From preceding formulas 01 and on can be shown to be related according to the transcendental equation tan pi) /q)i: /0i tan (01/2) (22) This gives an implicit interrelation among the four frequencies fa, f1, fi and f2 from which any one can be obtained numerically when the other three are known. Thus when fc, f1, and f2, and hence 01, are specified, (pi is fixed; whence also The two design constants, given in terms of (pi and ,fiq/fc for reasons stated later, are

and the width 11 of the aperture may be found from the relationship d 2 l g a 1 tan V0.

A simple correction for the thickness of the partition 51, 52 can be made, if required, from measured curves of the ratio ---a$i/)\g:1/gi plotted against d/a for various thicknesses.

At the quarter-wave frequency fiq the normalized image impedance ZUiq for the mid-guide sec tion becomes and the corresponding impedance wi for the mid-iris section is Since 0 5i 1r/2 i 1r, the former impedance is greater than, and the latter obviously less than, unity. The narrower the band, the greater the departure of each from unity. As gel is obtainable simply from either image impedance, in; and either wiq or wi can serve as another pair of parameters from which to determine g1 and 11. Such a pair plays an important part in the invention when combining dissimilar sections.

Another important formula which will be used later is that for the slope of the phase constant, B'iq, at the frequency fiq, given by df) L a-M (31) where This slope is positive, which agrees with my phase constant theorem that the phase constant in a wave-filter always increases with frequency throughout each transmitting band, first stated on page 5 and proved on page 3'7 of my article entitled, Theory and design of uniform and composite electric wave-filters, published in the Bell System Technical Journal, January 1923.

At any frequency which involve only the ratios of frequencies to fc.

Wave-guide wave filter with capacitive irises A type of capacitive iris suitable for use in the networks of the invention is shown in Fig. 5. The rectangular wave guide 55 is cross-sectioned just ahead of the iris which is formed by the two halves 51, 58 of a thin, metallic, transverse partition with a central aperture 59 therebetween. The aperture 59 extends all the way between the narrower sides of the guide 56 in the direction of the dimensions a and-has a width h in the direction of the electric field E. The capacitive iris may, of course, take any other appropriate physical form.

Proceeding in the same manner as above, design formulas for a filter section using capacitive irises will now be presented. The subscript used with the symbols 3, g, X, 6, l, B, w, and w indicates that the symbol applies to a capacitiveiris filter section.

The normalized susceptance So of a thin capacitive iris of the type shown in Fig. is positive and inversely proportional to the guide wavelength, A Put where 9c is a positive constant which fixes the magnitude of the susceptance. Then it 1 g lc X==- 36 t 2 f/frfc/f which approaches the reactance of a strict capacity as the ratio f/fc increases above unity. It follows that k war- 8 at the quarterwave frequency fcq,

fiqZc=1r/25cq/2 (40 Bcq=1r/2 (41) and at f2.

fl2Zc=1r/25c2 (42) 30 :0 (43) Define here for use with capacitive irises the two radian angles The angles 0c and (pe are related according to the transcendental equation goc tan goc=0c/tan (6c/2) (4'7) which interrelates the four frequencies fc, f1, fcq

and f2. Hence & 2 f. ft (1) 1 (48) The two design constants are given by h=tan (a- WUMflV- and i A comparison of Figs. 4 and 6 readily shows that for the same transmitting band the spacing of the capacitive irises must be greater than that of the inductive irises, their ratio being 5:55: (fl/ra Corrections for thickness of the capacitive iris can be made, if necessary.

At the quarter-wave frequency fc the normalized image impedance wcq for the mid-guide section is and the corresponding impedance w'cq for the since 1r/2 6cq 0 and 1r zpc 31r/2.

The slope of the phase constant B'cq at the frequency jcq is I wish to point out here that for the same fc and fiq=fcq the co'efilcients of D1 and Dc in Formulas 31 and 5'7, respectively, are identical and independent of band width for all inductive and capacitive type sections. This common property simplifies the application of these formulas where the ratios of phase slopes appear.

Wave-guide impedance transformers It will be seen from Formulas 29, 30 and 55, 56 that the image impedances of the wave-guide wave-filter sections described above, both inductive and capacitive, differ from the characteristic impedance is of the associated wave guide at frequencies near the center of their transmitting bands. 1 These departures increase as the band widths are narrowed.

Any such section having a real normalized image impedance, w, can be connected to the wave guide of normalized characteristic impedance, 1, without impedance irregularity by the wellknown method of inserting between them a single non-dissipative quarter-wave line or symmetrical section whose impedance at this frequency is the geometric mean of the two impedances to be connected, that is, /20. Then the impedances in the two directions at each junction are matched, thus allowing maximum transfer of energy across the junctions. However, these impedances match only at the one frequency, becoming more or less mismatched at frequencies departing from it. Such mismatches produce reflection losses and a narrow transmitting band as in a simple resonant circuit. This l-section method of impedance matching is therefore usually inadequate for matching over a frequency range. To accomplish the latter more satisfactorily a more complex structure is required.

To fill this need the present invention, therefore, presents dissymmetrical wave-guide structures which function as impedance-transforming wave-guide wave filters. The method of design is based upon the use of image parameters which I introduced on page 611 of my article entitled Transmission characteristics of electric wavefilters, published in the Bell System Technical Journal, October 1924. Here the transformer is designed so that its image impedances match the terminal impedances, assumed to be real, exactly at a single frequency near the center of the desired transmitting band. These image impedances, which are here real and vary slowly with frequency, continue to match the terminal impedances approximately over an appreciable frequency band, which results in good transmission through the transformer.

The general transformer in accordance with the invention is made up of n dissimilar sections, of the types described above, connected in tandem. All sections have identical associated wave guides, a property which facilitates their physical merger into a composite structure. For example, Fig. 7 shows schematically an embodiment in which 11:2. The transformer comprises two dissimilar mid-guide sections of the type shown in Fig. 1 connected in tandem between a first pair of terminals 24, 25 and a second pair 26, 21. The first section consists of a wave guide of length 211 with a shunt branch 3! of impedance iXr connected at the mid-point. The second section is made up of a wave guide of length 212 with a central shunt branch 38 of impedance 2'X2. The transformer has prescribed real image impedances We. at the terminals 24, 25 and Wb at the terminals 26, 21 at a specified frequency near the center of a transmitting band. To ensure the location of such a transmitting band, all sections are required to have the same quarter-wave frequency, fq=fiq=fcq, equal to this preassigned frequency, irrespective of the widths of their individual transmitting bands. Since each of the n sections has two parameters, a total of 2n relations is necessary to fix them. Such general relations which apply to the n-section transformer at the preassigned frequency, fq, are here taken to be the following (omitting from the T's, ws etc., the subscripts, i, z'q, etc., and the exponent prime and adding the subscript numbers, 1, 2, n, to identify the sections) T1=ii1r/2, Tn ii1r/2 (61) lp 2 p,, (62) where m, @n-Z are arbitrary positive parameters which relate the ratios of successive image impedances to the ratio of the impedances of the first two sections nearest terminals 24, 25,

aya: a a/ and Here ma, ya and we are, respectively, the normalized open-circuit, short-circuit and image impedances of the transformer at terminals 24, 25 of section 1, while :cb, 11b and we are the corresponding ones at terminals 25, 21 of section n. It will be seen that Formulas 61 to 64 contain the necessary 2n relations.

From the impedance-invention property of a succession of quarter-Wave sections satisfying 61 (see Formula 67) we have a simple relation between the two image impedances of the transformer in terms of those of the 71 sections, namely if n is even,

where 2 is the normalized input impedance of a section when terminated by the normalized impedance 25t- This is to be applied successively at each junction, beginning at the nth section. Although these expressions become more and more complicated as n increases, their product,

ways, simplifies at the quarter-wave frequency, f where for each section tanh T ii but it contains various ratios of these infinite hyperbolic tangents, which are of indeterminate form. The latter can, however, be evaluated readily by differentiating numerator and denominator to give, for example,

tanh T tan B B'g tanh T tan B B by Formulas 31, 57 or both, depending upon the types of iris. That is, as the frequency approaches ,f B1 and B2 each approach :1r/2 by assumption in (61) but not at the same rate, and although the tangent of each approaches infinity, the ratio of these tangents approaches a finite positive value given by (68).

To illustrate the above procedure, general formulas will now be obtained for 2-section, 3-section, and 4-section transformers. Similar formulas can be derived for cases where n is greater than 4.

There will first be considered the 2-section transformer, one embodiment of which is shown in Fig. 7 and others in Figs. 8 and 13. The structure shown schematically in Fig. 8 comprises two dissimilar mid-iris sections of the type shown in Fig. 2 connected in tandem between the terminals 24, 25 at one end and the terminals 26, 21 at the other end. The first section consists of a Wave guide of length 2Z1 with a shunt branch 39 of impedance i2X1 at the left end and a similar branch at the other end. The second section is made up of a wave guide of length 212 with a shunt branch 4| of impedance i2X2 at the right end and a similar branch at the other end. The two central shunt branches are merged into a single branch 40 of impedance i2X1X2/(X1-l-Xz). In the transformer of Fig. 13, the first section of Fig. 7 and the second section of Fig. 8 are connected in tandem to form a mixed combination.

For a 2-section transformer, two applications of (6'7) for and two for ya give Limit at =-g-f as) and At the design frequency, fq, their product reduces to Combining these with (68) we find the fundamental formulas for two sections to be It will be seen from Formulas 29, 30, 32, 55, 56, I

D2 are functions only of the radian angles, p1 and (p2, respectively, of the two sections. The pair of simultaneous Equations '73 and '74, subject to 75, are suflicient to determine (p1 and 2. Because of the transcendental nature of these formulas, more explicit ones for (p1 and (p2 cannot be given, but for any physical numerical values of we and wt, they can be solved readily by a method of successive approximations of U2, beginning with U2=1.

These values of the s, together with the preassigned value of fq/fc, fix the design constant g and the length 21 of each of the transformer sections. If the section has an inductive iris, Equations 24 and 25 are used. If the iris is capacitive, Equations 49 and 50 are applicable.

Figs. 9, 10, 14, and 15 show schematically four different 3-section transformers in accordance with the invention. In Fig. 9, all of the sections are of the mid-guide type, and in Fig. 10 they are all of the mid-iris type. The structure of Fig. 14 is made up of two mid-guide sections and one mid-iris, and Fig. 15 of one mid-guide and two mid-iris. Fig. 9 is obtained by adding to Fig. 7 a third section of length 2Z3 with a branch 42 of impedance iXs connected in shunt at the midpoint. Fig. 10 is derived from Fig. 8 by adding a section of length 213 with a shunt branch 44 of impedance i2X3 at the right end and a similar branch at the other end. The two parallel shunt branches at the junction of sections 2 and 3 are merged to form the branch 43 having an impedance of iZXzXa/(Xz-l-Xa). In Fig. 14 a midiris section 3 of the type shown in Fig. 10 is added to the two mid-guide sections of Fig. 7. In Fig. 15 a similar mid-iris section 3 is added to the network of Fig. 13.

Design formulas for a B-section transformer, examples of which are shown in Figs. 9, 10, 14, and 15, are

where A numerical solution for (pr, (p2 and as is here also obtained by successive approximations, beginning with U3=1.

Figs. 11, 12, and 16 show schematically three different 4-section transformers in accordance with the invention. In Fig. 11 all of the sections are of the mid-guide type, in Fig. 12, they are of the mid-iris type, and in Fig. 16 two are midguide and two are mid-iris. Fig. 11 is obtained by adding to Fig. 9 a fourth section of length 224. with a branch 45 of impedance iX4 connected in shunt at the mid-point. Fig. 12 is derived from Fig. 10 by adding a section of length 214 with a shunt branch 4'! of impedance i2X4 at the right end and a similar branch at the left end. The two parallel shunt branches at the junction of sections 3 and 4 are merged to form the branch 46 whose impedance is i2X3X4/(X3X4). In Fig. 16 the transformer of Fig. 7 is substituted for the first two mid-iris sections of Fig. 12. Here the branch 44 has an impedance of iZXa, the same as that of the like-designated branch in Fig. 10.

have the design constants g1 tandagc' or :the two sections. In this case we cannormalize (85) and obtain by (12') and (37 the normalized susceptance s" of therequired iris. .It is i k i This can be simulated in any particular case .Toobtain 1, c2, rps, and s a in; a numerical. case we again. use. successive. approximations and start with. U4=1. The convergence to a solution is rapid.

These illustrative formulas indicate a tapering of the magnitudes of the section impedances between the network image impedances we and we. Hence, when applying them some care is required inchoosing. the kinds of section, depending upon theemagnitudes of 7172. and an, to ensure a physical solution with positive values of the We. A choice for each of the three cases arising is as follows:

Where 1wa wb, use mid-guide sections as in 'Figs. '7. 9 and 11.

Where 1wa wa use mid-iris sections, as in Flgs48, .10 and 12.

'Where wa 1 wz use mid-guide sections be ginning at terminals 24, 25 and mid-iris ending at terminals 26, 27 in the approximate proportion of we to wb, as in Figs. 13, 14, and 16.

The parameters 231, pa, etc., appearing in the formulas can be chosenso as to alter somewhat the width of the resulting transmitting band of the transformer and its impedance characteristics. Values of unity normally givegood results.

An alternative to an extension of the above sets of formulas to larger values of n would be the division of the total impedance range, we to 1121), into two or more ranges and then a separate design for each range with. any of the formulas already derived. The number of sections chosen for each range. could be roughly proportional to the impedance ratio required. The division points might be taken, for example, so as to give the same transformer ratios for all. It is" obvious that when these separate structures corresponding to the diflerentimpedancelevels are connected in tandem they are joined on an image impedance basis and the'tot'al structure will have the desired image impedances, we and we, at its terminals.

Where two dissimilar mid-iris sections are joined together there is a single merged impedance-such as 40 in Fig. 8, for example. This has a 'rea'ctance X given by which must be simulated by the proper iris. If 'both sections have inductive irises, X will be inductive; so from (11) and (85) the design constant 91' for the corresponding inductive iris is If both sections have capacitive irises, the design constant go for the corresponding merged capacitive iris is by (36) and (85) If, however, one section is inductive and the other capacitive, 'X' will .be anti-resonant and over the important frequency range by an antiresonant iris having two parameters such as, for example, one having a central rectangular opening in the iris whose sides are parallel to those of the wave guide. (See page 121 of "Microwave Transmission Design Data, published by the Sperry Gyroscope Company, May 1944.) Fitting this at the design frequency IQ and at h orfz will be satisfactory.

An approximation can be made to (88) by fitting its value s at fq with an inductive iris when s is negative, and with a capacitive iris when s is positive. In the former case we would have from (10) i'=2/ (f./f. -1 (s9) and in the latter case from (35) se m (nucl us; (9

To exemplify the present invention, a specific design will here be outlined for a wave-guide wave filter together with a wave-guide transformer suitablefor interconnecting the former to a wave guide having the same transverse dimensions.

The composite structure is shown schematically in Fig. 17. All of the sections are of the midguide type (Fig. 1) and have inductive irises such as the one shown in 'Fig. 3. The transformer, between terminals 24, 25 and 26, 21, has two sections. It is a special case of the circuit of Fig. 7 with the general shunt impedance branches 3'! and 38 replaced, respectively, by the inductances 3i and 32. Only one filter section is shown, between terminals 26, 2'1 and 28, 29, but it will be understood that more sections may be added, if required. It consists of a wave guide of length 2Zi with a centrally-positioned shunt inductance 35). Since this filter section is of the mid-guide type, it is preferable that the transformer associated therewith terminate in a mid-guide section on the end connected to the filter. The transformers shown in Figs. 9, 11, 13, 14, 15, and 16 all have a mid-guide termination on at least one endand may, therefore, be sub stituted for the transformer shown in Fig. 17. On the other hand, if the filter terminates in a mid-iris section (Fig. 2) the transformer should have a mid-iris termination at the. junction with the filter. Suitable transformers are shown in Figs. 8, 10, 12, 13, 14, 15, and 16.

It will be assumed that the component wave guide is rectangular in cross section and has an inside width a of 4.758 centimeters (1.873 inches) and an inside height b of 2.215 centimeters (0.872 inch). From Equation 1 the critical frequency'of the guide is #:3152613 megacycles per second Its normalized characteristic impedance we is, of course, unity at all frequencies.

It will be assumed that the filter is to have the following preassigned transmitting band limits, in megacycles per second:

and

From (19), (2-2) and (23) we have 61:2.85684 radians, -2.99258 radians and fi ==4046.08 megacycles per second. Substitution of these values in (24) and (25) gives the design constants and Zi:1.1090 inches From (28) we obtain d/a 0.2596, which when corrected from experimental curves for the thickness of the iris to be used (0.045 inch) gives d/a:0.291 and 01:05:45 inch for the iris represented by the inductance 39. The normalized mid-guide image impedance at fi becomes from (29) wi :13.3968. All of the dimensions required for the construction of the filter section have thus been determined.

Fig. 18 gives certain impedance-frequency characteristics of interest. They are shown only in the transmitting band, between the cut-off frequencies f1 and f2. It should be noted that, as indicated, there is a break in the ordinate scale between the point 3 shown at the left, and the point Hi shown at the right. Curve Ed shows the normalized mid-guide image impedance tor of the filter section at terminals 26, 2'! or 28, 29 of Fig. 1'7, found from Formula 7 used in conjunction with 33 and 34. The normalized characteristic impedance 717g of the wave guide, which is unity at all frequencies, is shown by curve 65. The other characteristics will be discussed later.

The Z-section transformer shown in Fig. 1'7,

between terminals 24, 25 and 26, 21, is to be de l signed at the frequency f fi so as to connect with the wave guide at terminals 25, 25 and the above wave filter at terminals 26, 2? without impedance irregularities. Hence, in addition to the values of fc and ft; above, the transformer data at f are wa:w :1.() and.

wb:wi :13.3968

Applying Formulas 73 to 75, together with 29 and 32, for each section, we obtain the solution where Uz:1.38729. From (24) and (25) the following design constants are obtained:

gi:1.0917 92:0.25866 11 03972 inch Z2:1.0883 inches tandem combination of the two transformer sections, at the terminals 24, 25, is given by curve 68. This characteristic was computed in the same way as curves E4, 65 and El, with the aid also of Formulas 5, 63, 69 and '70. Curve 69 gives the normalized image impedance wt for the transformer at the terminals 25, 21. This characteristic was computed by means of formulas similar to those used for curve 68 but with the subscripts a and, b, 1 and 2 interchanged. These various characteristics may be readily compared in Fig. 18. In particular, it is to be noted that 10a not only equals w at the quarter-wave frequency ji of the wave filter but approximates it well over a considerable frequency range on either side. Aso, the limits of' the transmitting band of the transformer, itself a wave filter, occur where wuzo; here they are just within the filter band f1, f2. Likewise, 'LUb equals an at fi and approximates it very well over the band. Hence, when these structures are connected in tandem, as shown in Fig. 17, there will be appreciable impedance irregularities only at the edges of the band, which would there cause minor ripples in an over-all transmission characteristic.

It is obvious also from Fig. 18 that in this example the curve 68 could be made to cross the curve 65 at two frequencies, one on either side or" fi by choosing at fiq a design value, wa w :1.0. Similarly, the curve 69 could also be made to cross the curve 64 at two frequencies by taking a design value, wb wiq. There would then be some impedance mismatches at the frequency fiq. The above serves to indicate the flexibility of this method of design.

Fig. 19 shows schematically another composite network in accordance with the invention. It consists of two networks of the type shown in Fig. 17, connected back-to-back, with two additional mid-guide filter sections connected in tandem between them. Thus, in Fig. 19 the Z-section transformer at the left end between terininals 24, 25 and 28, 21 is identical with the transformer at the right end between terminals 24, 25 and 26, 21, and each is the same as the one shown in Fig. 17 between terminals 24, 25 and 2t, 2?. The four mid-guide filter sections between terminals 26, 21 and 26, 21 are all alike and each has a centrally-positioned shunt inductance 30. The transformers serve to match the image impedance of the filter sections to the characteristic impedance of the wave guide to which the network is connected at each end.

A physical embodiment of the network of Fig. 19 is shown in longitudinal section in Fig. 20 and in transverse section in Fig. 21. The component wave guide is rectangular in cross section and made in three sections H, 12 and 13, connected in tandem by means of the end coupling flanges 34 and associated screws, not shown. The sections of guides H and 13 are associated, respectively, with the end transformers, and the interposed section 12 with the filter. The irises are of the inductive type shown in Fig. 3 and each is formed by the two halves M, 15 of a thin, metallic, transverse partition with a central aperture therebetween extending all the way between the wider sides of the guide. The irises 30, 3| and 32 correspond, respectively, to the inductances 30, 3| and 32 shown in Fig. 19. Four trimming screws 33 are provided for tuning the cavities with which they are associated, to adjust the transmission characteristic of the network, if desired. More trimming screws may, of course, be provided.

In Fig. 22 curve 36 is the insertion loss-frequency characteristic of the structure shown in Figs. 20 and 21 measured between wave guides having the same cross-sectional dimensions as the wave-guide sections H, 12 and 13. The loss is small and nearly fiat throughout the transmission band and the cut-offs are quite steep. To get this characteristic required slight adjustments of the trimming screws 33 and the addition of thin shims at the two junctions 80 and 3! between the flanges 34. For comparison, curve 35 of Fig. 22 gives the insertion loss characteristic obtained when the terminal transformers are omitted from the network. The high loss over most of the theoretical band is due mainly to the mismatch between the image impedance of the filter sections and the characteristic impedance of the guide connected thereto. At the points l6, l1 and 18 the loss is low because, at these frequencies, the phase constant of the filter is equal to an integral multiple of 1r radians so that the filter becomes transparent, as shown by Formula 67. A comparison of the curves 35 and 36 shows the great improvement obtainable by terminating wave filters with impedance transformers designed in accordance with the present invention.

The transformers alone, connected back-toback, also have a useful band-pass insertion loss characteristic. In Fig. 23 the solid-line curve is a measured characteristic and the broken-line curve is the theoretical computed one. The two characteristics may be brought into even closer agreement by the addition and adjustment of a trimming screw at the junction of the transformers.

Any of the transformers disclosed herein may, of course, be connected back-to-back, without any intervening filter sections, to provide bandpass structures. For example, a it-section transformer of the type shown in Fig. 11 was designed with inductive irises for the same terminal i-m pedances rm and we as in the previous example of Fig. 17. It was first designed as two 2-section transformers in tandem which have the same impedance ratios and the same normalized image impedance, /wawb=3.66016, at their junction. The identical design was obtained later in one operation with Formulas 80 to 84 in which p1=pz=1.0, meaning that the ratios of successive impedances at fi are equal. Two of these e-section transformers were constructed and, without any adjustment, measured back-to-back between wave guides. The measured insertion loss characteristic was found to be similar to the solid-line curve shown in Fig. 23, but more selective, with band limits close to f1 and f2 as desired.

The insertion loss L in decibels for the measured characteristics shown herein was obtained at each frequency by determining the voltage standing-wave ratio M in decibels and applying the readily derived conversion formula L=2O 10510 cosh frequency fq and having unequal real image impedances which, when normalized with respect to an impedance is, are equal respectively to We and we at said frequency, said filter comprising three sections connected in tandem, each of said sections comprising a section of uniform wave guide of length 21!) having a characteristic impedance equal to It, a critical frequency fc, and a phase constant B at said frequency and a centrally positioned shunt inductor having a positive design constant on fixing the magnitude of its susceptance, and the nth section, where n is successively equal to 1, 2 and 3, having a normalized image impedance wn at each end, where where 01 is an arbitrary positive constant and w, m and we correspond respectively to the filter sections 1, 2 and 3,

wn=tan r n/2i and 2. A microwave filter adapted to transmit a band of frequencies approximately centered at a frequency j and having unequal real image impedances which, when normalized with respect to an impedance is, are equal respectively to we and we at said frequency, said filter comprising three sections connected in tandem, each of said sections comprising a section of uniform wave guide of length 2Zn having a characteristic impedance equal to la, a critical frequency f0, and a phase constant fiq at said frequency and. a centrally positioned shunt capacitor having a positive design constant gm fixing the magnitude of its suscept'ance, and the nth section, where n is successively equal to 1, 2 and 3, having a normalized image impedance wn at each end, where and the angles p1, p2 and (p3 corresponding respectively to the filter sections 1, 2 and 3 satisfy at said frequency the following equations:

where 171 is an arbitrary positive constant and an, 102 and we correspond respectively to the filter sections 1, 2 and 3,

wn=tan ((/m/2) and 3. A microwave filter adapted to transmit a band of frequencies approximately centered at a frequency f and having unequal real image impedances which, when normalized with respect to an impedance k, are equal respectively to we and we at said frequency, said filter comprising three sections connected in tandem, each of said sections comprising a section of uniform wave guide of length 2Zn having a characteristic impedance equal to k, a critical frequency fc, and a phase constant B at said frequency and at the respective ends thereof two shunt inductors each having a positive design constant gn fixing the magnitude of its susceptance, and the nth section, where n is successively equal to 1, 2 and 3, having a normalized image impedance wn at each end, where.

and the angles 1, 2 and. 03 corresponding respectively to the filter sections 1, 2 and 3 satisfy at said frequency the followin equations:

w1=waUa w2=p1Vm w3=p1wb/U3 U: i+(' 2/ 1) 2+P1 :i J i+( 1/ 2) 2+ a/P1 where 171 is an arbitrary positive constant and un, um and m correspond respectively to the filter sections 1. 2 and 3,

1011 5111 (p11 and P 1;g 5+ (s n 4. A microwave filter adapted to transmit a band of frequencies approximately centered at a frequency j and having unequal real image impedances which, when normalized with respect to an impedance lc, are equal respectively to we and we at said frequency, said filter comprising three sections connected in tandem, each of said sections comprising a section of uniform wave guide of length 2111 having a characteristic impedance equal to k, a critical frequency fc, and a phase constant S at said frequency and at the respective ends thereof two shunt capacitors each having a positive design constant gn fixing the magnitude of its susceptance, and the nth section, where n is successively equal to l, 2 and 3, having a normalized image impedance wn at each end, where ln gon/ 215g yn=tan (gin-1r) and the angles 01, (p2 and (p3 corresponding respectively to the filter sections 1, 2 and 3 satisfy at said frequency the fOlIOWil'ig equations:

where 121 is an arbitrary positive constant and 101, wz and L03 correspond respectively to the filter sections 1, 2 and 3,

wn=sin ((pn) and 5. A filter in accordance with claim 1 in which the constant 111 is approximately equal to unity.

6. A filter in accordance with claim 1 in which we is as least as large as unity, and 20b is larger than we.

7. A filter in accordance with claim 2 in which the constant 1 1 is approximately equal to unity.

8. A filter in accordance with claim 2 in which we is at least as large as unity, and wb is larger than 103.

9. A filter in accordance with claim 3 in which the constant in is approximately equal to unity.

10. A filter in accordance with claim 3 in which we is not larger than unity, and wt is smaller than we.

11. A filter in accordance with claim 4 in which the constant 121 is approximately equal to unity.

12. A filter in accordance with claim 4 in which we is not larger than unity, and we is smaller than we.

13. A microwave filter adapted to transmit a band of frequencies approximately centered at a frequency fq and having unequal real image impedances which, when normalized with respect to an impedance k, are equal respectively to we and we at 'said frequency, said filter comprising the tandem combination of two adjacent sections of the same type and a section of different type, each of said sections comprising a uniform wave guide of length 2Zn having a characteristic impedance equal to k, a critical frequency ,fc, and a phase constant flq at said frequency, one of said types of sections being of'a first type comprising a centrally positioned shunt inductor, the other of said types of sections being of a second type comprising two shunt inductors positioned at the respective ends thereof, each of said inductors having a positive design constant gn fixing the magnitude of its susceptance and the nth section, Where n is successively equal to 1, 2 and 3, having a normalized image impedance wn at each end, where for each of said sections and the angles i, p2 and gas corresponding respectively to the filter sections 1, 2 and 3 satisfy at said frequency the following equations:

where in is an arbitrary positive constant and wi, wz and 103 correspond respectively to the filter sections 1, 2 and 3, and

+ n (Wn' for said first type of section; and

wn=sin gun with claim 13 in 21 which the constant n is approximately equal to unity.

17. A filter in accordance with claim 13 in which we is larger than unity, and we is smaller than unity.

18. A microwave filter adapted to transmit a band of frequencies approximately centered at a frequency fq and having unequal real image impedances which, when normalized with respect to an impedance is, are equal respectively to we and we at said frequency, said filter comprising the tandem combination of two adjacent sections of the same type and a section of different type, each of said sections comprising a uniform wave guide of length 2Zn having a characteristic impedance equal to k, a critical frequency fc, and a phase constant c at said frequency, on of said types of sections being of a first type comprising a centrally positioned shunt capacitor, the other of said types of sections being of a second type comprising two shunt capacitors positioned at the respective ends thereof, each of said capacitors having a positive design constant on fixing the magnitude of its susceptance, and the nth section, where n is successively equal to 1, 2 and pmzawb/ o lwa B2me wear want Z1z: pn/2,Bq n=en w-a N h/T 2i and the angles (p1, 2, s and 4 corresponding respectively to the filter sections .1, 2, 3 and 4 satisfy at said frequency the following equations:

3, having a normalized image impedance 'LUn at where :01 and 322 are arbitrary positive constants each end, where for each of said sections zn==0n/2fiq n= (awn W and the angles (p1, (p2 and ya corresponding respectively to the filter sections 1, 2 and 3 satisfy at said frequency the following equations:

where 321 is an arbitrary positive constant and wt, 102 and we correspond respectively to the filter sections 1, 2 and 3, and

for said second type of section.

'19. A filter in accordance with claim 18 in which said two adjacent sections are of said first type.

20. A filter in accordance with claim 18 in which said two adjacent sections are of said second type.

21. A filter in accordance with claim 18 in which the constant n is approximately equal to unity.

22. A filter in accordance with claim 18 in which we is larger than unity, and we is smaller than unity.

23. A microwave filter adapted to transmit a hand of frequencies approximately centered at a frequency fq and having unequal real image impedances which, when normalized with respect to and 101, we, we and W4 correspond respectively to the filter sections 1, 2, 3 and 4,

24. A filter in accordance with claim 23 in which the constants n and 102 are each approximately equal to unity.

25. A filter in accordance with claim 23 in which 'lUa is at least as large as unity, and w is larger than we.

26. A microwave filter adapted to transmit a band of frequencies approximately centered at a frequency j and having unequal real image impedances which, when normalized with respect to an impedance is, are equal respectively to we and we at said frequency, said filter comprising four sections connected in tandem, each of said sections comprising a uniform wave guide of length 21!} having a characteristic impedance equal to k, a critical frequency fc, and a phase constant flq at said frequency and a centrally positioned shunt capacitor having a positive design constant on fixing the magnitude of its susceptance, and the nth section, where n is successively equal to 1, 2, 3 and 4, having a normalized image impedance wn at each end, where gnztaln (P11 ")v (fq/fc) and theangles 1, 2, es and 4 corresponding respectlvely to the filter sections 1, 2, 3 and 4 satisfy at said frequency the following equations:

w1=p1 p2- zua wb U4 w2=p1 p2 wa wb Us w3=p1 p2 wd wb U4 and w4=pr pz *wa wu U4 where 23" where 121 and 102 are arbitrary positive constants and an, wz, wa and 204 correspond respectively to the filter sections 1, 2, 3 and 4,

27. A filter in accordance with claim 26 in which the constants p1 and 102 are each approximately equal to unity.

28. A filter in accordance with claim 26 in which we. is at least as large as unity, and 'LUb is larger than w.

29. A microwave filter adapted to transmit a band of frequencies approximately centered at a frequency f and having unequal real image impedances which, when normalized with respect to an impedance k, are equal respectively to we and we at said frequency, said filter comprising four sections connected in tandem, each of said sections comprising a uniform wave guide of length 211:: having a characteristic impedance equal to k, a critical frequency fc, and a phase constant fl at said frequency and at the respective ends thereof two shunt inductors each having a positive design constant 911 fixing the magnitude of its susceptance, and the nth section, where n is successively equal to 1, 2, 3 and 4, having a normalized image impedance 71111 at each end, where an= cn)/w/(fq/fc) and the angles (p1, 2, s and (p4 corresponding respectively to the filter sections 1, 2, 3 and 4 satisfy at said frequency the following equations:

Zn=qJn/2flq n= n c")/x (fq/f) and the angles p1, (722, p3 and (p4 corresponding respectively to the filter sections 1, 2, 3 and 4 satisfy at said frequency the following equations:

where 101 and 122 are arbitrary positive constants and w, wz, 103 and L04 correspond respectively to the filter sections 1, 2, 3 and 4,

and

fi sin o y-710 (Pu 7r) 33. A filter in accordance with claim 32 in which the constants p1 and m are each approximately equal to unity.

34. A filter in accordance with claim 32 in which wa is not larger than unity, and we is smaller than we.

35. A microwave filter adapted to transmit a band of frequencies approximately centered at a frequency f and having unequal real image impedances which, when normalized with respect to an impedance is, are equal respectively to we and we at said frequency, said filter comprising four sections connected in tandem, each of said sections comprising a uniform wave guide of length 2Zn having a characteristic impedance where 171 and p2 are arbitrary positive constants and 7.01, 202, ws and wr correspond respectively to the filter sections 1, 2, 3 and 4,

30. A filter in accordance with claim 29 in which the constant 291 is approximately equal to unity.

31. A filter in accordance with claim 29 in which we is not largerthan unity, and we is smaller than we.

32. A microwave filter adapted to transmit a band of frequencie approximately centered at a frequency f and having unequal real image impedances which, when normalized with respect to an impedance 7c, are equal respectively to we and we at said frequency, said filter comprising four sections connected in tandem, each of said sections comprising a uniform wave guide of length 2111 having a characteristic impedance equal to k,

equal to 10, a critical frequency fc, and a phase constant q at said frequency, two adjacent sections being of a first type comprising a centrally positioned shunt inductor, two adjacent sections being of a second type comprising two shunt inductors positioned at the respective ends thereof, each of said inductors having a positive design constant gn fixing the magnitude of its susceptance, and the nth section, where n is successively equal to 1, 2, 3 and 4, having a normalized image impedance wn at each end, where for each of said sections and th angles (p1, cpz, (p3 and (p4 corresponding respectively to the filter sections 1, 2, 3 and 4 satisfy at said frequency the following equations:

and and the angles (P1, 992, i s and (p4 corresponding m p 1/2p 1/4w w U respectively to the filter sections 1, 2;3 and 4 where where 101 and 112 are arbitrary positive constants satisfy at said frequency the following equations: and 101, wz, w: and 04 correspond respectively to 7/8 1/8 the filter sections 1, 2, 3 and 4, and

2 and 201, we, we and we correspond respectively w":m to the filter sections 1, 2, 3 and 4, and

for said second type of section. I

36. A filter in accordance with claim 35 in which the constants in and m are each approxm ;;f;+ (ar imately equal to unity.

37. A filter in accordance with claim 35 in wnztan qm/2) which 10a is larger than unity, and wt is smaller than u yfor said first type of section; and

38. A microwav filter adapted to transmit a band of frequencies approximately centered at wn=sin a frequency ,f and having unequal real image impedances which, when normalized with refo said Second type of section spect to an impedance k, are equal respective- A filter-in accordance with claims; in

y to e a at S f q filter which the constants in and m are each approxicomprismg four sections connected in tandem, mately equal'to unity. 1

ch f said sections comprising a q a 40. A filter in accordance with claim 38 in uide o length 2 11 having a characterlstm which we is larger than unity, and we is smaller pedance equal to k, a critical frequency .fe, and than unity r a phas constant B at said frequency, two ad- OTTO ZOBEL Jaoent sections being of a first type comprising 40 a centrally positioned shunt capacitor, two adjacent sections being of a second type compris- REFERENCES CITED ing two shunt capacitors positioned at the re- The following references are of record in the spective ends thereof, each of said capacitors file of this patent: 5-

having a positive design constant gm fixing the 4 UNITED STATES PATENT magnitude of its susceptance, and the nth section, where n is successively equal to 1, 2, 3 and Number Name Date 4, having a normalized image impedance we at 2396,04; FOX 194.5

each end, where for each of said sections 2,432,093 FOX 1 Zn qm/2pa an 4 ,48 Mumford Feb. 6, 1951 

